Boltzmann Limit and Quasifreeness for a Homogenous Fermi Gas in a Weakly Disordered Random Medium

Abstract

We discuss some basic aspects of the dynamics of a homogenous Fermi gas in a weak random potential, under negligence of the particle pair interactions. We derive the kinetic scaling limit for the momentum distribution function with a translation invariant initial state and prove that it is determined by a linear Boltzmann equation. Moreover, we prove that if the initial state is quasifree, then the time evolved state, averaged over the randomness, has a quasifree kinetic limit. We show that the momentum distributions determined by the Gibbs states of a free fermion field are stationary solutions of the linear Boltzmann equation; this includes the limit of zero temperature.

Boltzmann Limit for a Homogeneous Fermi Gas with Dynamical Hartree-Fock Interactions in a Random Medium

“…The first term on the right hand side of (5.37) bounds the contribution from all basic ladder diagrams contained in the Duhamel expanded remainder term. A detailed discussion of an analogous result, and more details, can be found in [8,9,10,17]. 5.1.3.…”

“…This result remains valid in the regime λ = o(η 2 ). In the case λ = 0, it reduces to the one proven in [10].…”

“…To organize the terms in these sums of products of pair correlations, we introduce Feynman graphs. The set of Feynman graphs Γ n, n , withn = n + n ∈ 2N, is given as follows; see also [8,10,17]: -We consider two horizontal solid lines, which we refer to as particle lines, joined by a distinguished vertex which we refer to as the ρ 0 -vertex (corresponding to the term ρ 0 ( a + un a un+1 ). See Figure 1 for an example.…”

“…We point out that in our approach, the estimates provided in Lemmata 5.1 and 5.2, and in Section 6.5, require first integrating out all time variables s j , and subsequently integrating out the momenta u j in (4.42) and (4.37). This makes it possible to straightforwardly adopt estimates on crossing, nesting, and remainder terms from [8,9,10,17]. Accordingly, our discussion in this section can be kept short.…”

“…where 2n = n + n ∈ 2N. The thermodynamic limit L → ∞ and the upper bound in (5.31) are obtained in the same manner as in [8,9,10,17] where we refer for details. We do not repeat the discussion here, and instead refer to those references.…”